CYK Algorithm for Parsing General Context-Free Grammars

Why Parse General Grammars• Can be difficult or impossible to make

grammar unambiguous– thus LL(k) and LR(k) methods cannot work,

for such ambiguous grammars• Some inputs are more complex than simple

programming languages– mathematical formulas:

x = y /\ z ? (x=y) /\ z x = (y /\ z)– natural language:

I saw the man with the telescope.– future programming languages

Ambiguity

I saw the man with the telescope.

1)

2)

CYK Parsing AlgorithmC:John Cocke and Jacob T. Schwartz (1970). Programming languages and their compilers: Preliminary notes. Technical report, Courant Institute of Mathematical Sciences, New York University.

Y:Daniel H. Younger (1967). Recognition and parsing of context-free languages in time n3. Information and Control 10(2): 189–208.

K:T. Kasami (1965). An efficient recognition and syntax-analysis algorithm for context-free languages. Scientific report AFCRL-65-758, Air Force Cambridge Research Lab, Bedford, MA.

Two Steps in the Algorithm

1) Transform grammar to normal formcalled Chomsky Normal Form

(Noam Chomsky, mathematical linguist)

2) Parse input using transformed grammardynamic programming algorithm

“a method for solving complex problems by breaking them down into simpler steps. It is applicable to problems exhibiting the properties of overlapping subproblems” (>WP)

Balanced Parentheses Grammar

Original grammar GS “” | ( S ) | S S

Modified grammar in Chomsky Normal Form:S “” | S’

S’ N( NS) | N( N) | S’ S’ NS) S’ N)

N( (N) )

• Terminals: ( ) Nonterminals: S S’ NS) N) N(

Idea How We Obtained the Grammar

S ( S )

S’ N( NS) | N( N)

N( (

NS) S’ N)

N) )Chomsky Normal Form transformation can be done fully mechanically

Dynamic Programming to Parse Input

Assume Chomsky Normal Form, 3 types of rules:S “” | S’ (only for the start non-

terminal)Nj t (names for terminals)Ni Nj Nk (just 2 non-terminals on RHS)

Decomposing long input:

find all ways to parse substrings of length 1,2,3,…

( ( ( ) ( ) ) ( ) ) ( ( ) )

Ni

Nj Nk

Parsing an InputS’ N( NS) | N( N) | S’ S’ NS) S’ N)

N( (N) )

N( N( N) N( N) N( N) N)1

2

3

4

5

6

7ambiguity

( ( ) ( ) ( ) )

Parsing an InputS’ N( NS) | N( N) | S’ S’ NS) S’ N)

N( (N) )

N( N( N) N( N) N( N) N)1

2

3

4

5

6

7ambiguity

( ( ) ( ) ( ) )

( ( ) ( ) ( ) )

Algorithm IdeaS’ S’ S’

1 N( N( N) N( N) N( N) N)

2

3

4

5

6

7wpq – substring from p to qdpq – all non-terminals that could expand to wpq

Initially dpp has Nw(p,p)

key step of the algorithm:

if X Y Z is a rule, Y is in dp r , and Z is in d(r+1)q

then put X into dpq

(p r < q), in increasing value of (q-p)

AlgorithmINPUT: grammar G in Chomsky normal form word w to parse using GOUTPUT: true iff (w in L(G)) N = |w| var d : Array[N][N] for p = 1 to N { d(p)(p) = {X | G contains X->w(p)} for q in {p + 1 .. N} d(p)(q) = {} } for k = 2 to N // substring length for p = 0 to N-k // initial position for j = 1 to k-1 // length of first half val r = p+j-1; val q = p+k-1; for (X::=Y Z) in G if Y in d(p)(r) and Z in d(r+1)(q) d(p)(q) = d(p)(q) union {X} return S in d(0)(N-1) ( ( ) ( ) ( ) )

What is the running time as a function of grammar size and the size of input?

O( )

Parsing another InputS’ N( NS) | N( N) | S’ S’ NS) S’ N)

N( (N) )

( ) ( ) ( ) ( )

N( N) N( N) N( N) N( N)1

2

3

4

5

6

7

Number of Parse Trees

• Let w denote word ()()()– it has two parse trees

• Give a lower bound on number of parse trees of the word wn

(n is positive integer)w5 is the word

()()() ()()() ()()() ()()() ()()()

• CYK represents all parse trees compactly– can re-run algorithm to extract first parse tree, or

enumerate parse trees one by one

Transforming to Chomsky Form

• Steps:1. remove unproductive symbols2. remove unreachable symbols3. remove epsilons (no non-start nullable symbols)4. remove single non-terminal productions X::=Y5. transform productions of arity more than two6. make terminals occur alone on right-hand side

1) Unproductive non-terminals

What is funny about this grammar: stmt ::= identifier := identifier | while (expr) stmt | if (expr) stmt else stmt expr ::= term + term | term – term term ::= factor * factor factor ::= ( expr )

There is no derivation of a sequence of tokens from exprWhy? In every step will have at least one expr, term, or factorIf it cannot derive sequence of tokens we call it unproductive

How to compute them?

1) Unproductive non-terminals• Productive symbols are obtained using these

two rules (what remains is unproductive)– Terminals are productive– If X::= s1 s2 … sn is rule and each si is productive

then X is productive

stmt ::= identifier := identifier | while (expr) stmt | if (expr) stmt else stmt expr ::= term + term | term – term term ::= factor * factor factor ::= ( expr ) program ::= stmt | stmt program

Delete unproductivesymbols.

Will the meaning of top-level symbol (program) change?

2) Unreachable non-terminals

What is funny about this grammar with starting terminal ‘program’ program ::= stmt | stmt program stmt ::= assignment | whileStmt assignment ::= expr = expr ifStmt ::= if (expr) stmt else stmt whileStmt ::= while (expr) stmt expr ::= identifierNo way to reach symbol ‘ifStmt’ from ‘program’What is the general algorithm?

2) Unreachable non-terminals

• Reachable terminals are obtained using the following rules (the rest are unreachable)– starting non-terminal is reachable (program)– If X::= s1 s2 … sn is rule and X is reachable then

each non-terminal among s1 s2 … sn is reachable

Delete unreachablesymbols.

Will the meaning of top-level symbol (program) change?

3) Removing Empty Strings

Ensure only top-level symbol can be nullable

program ::= stmtSeq stmtSeq ::= stmt | stmt ; stmtSeq stmt ::= “” | assignment | whileStmt | blockStmt blockStmt ::= { stmtSeq } assignment ::= expr = expr whileStmt ::= while (expr) stmt expr ::= identifier

How to do it in this example?

3) Removing Empty Strings - Result

program ::= “” | stmtSeq stmtSeq ::= stmt| stmt ; stmtSeq | | ; stmtSeq | stmt ; | ; stmt ::= assignment | whileStmt | blockStmt blockStmt ::= { stmtSeq } | { } assignment ::= expr = expr whileStmt ::= while (expr) stmt whileStmt ::= while (expr) expr ::= identifier

3) Removing Empty Strings - Algorithm

• Compute the set of nullable non-terminals• Add extra rules

– If X::= s1 s2 … sn is rule then add new rules of form X::= r1 r2 … rn

where ri is either si or, if

• Remove all empty right-hand sides• If starting symbol S was nullable, then

introduce a new start symbol S’ instead, and add rule S’ ::= S | “”

si is nullable thenri can also be the empty string (so it disappears)

3) Removing Empty Strings

• Since stmtSeq is nullable, the rule blockStmt ::= { stmtSeq }gives blockStmt ::= { stmtSeq } | { }

• Since stmtSeq and stmt are nullable, the rule stmtSeq ::= stmt | stmt ; stmtSeqgives stmtSeq ::= stmt | stmt ; stmtSeq

| ; stmtSeq | stmt ; | ;

4) Eliminating single productions

• Single production is of the formX ::=Y

where X,Y are non-terminals program ::= stmtSeq stmtSeq ::= stmt | stmt ; stmtSeq stmt ::= assignment | whileStmt assignment ::= expr = expr whileStmt ::= while (expr) stmt

4) Eliminate single productions - Result

• Generalizes removal of epsilon transitions from non-deterministic automata

program ::= expr = expr | while (expr) stmt | stmt ; stmtSeq stmtSeq ::= expr = expr | while (expr) stmt | stmt ; stmtSeq stmt ::= expr = expr | while (expr) stmt assignment ::= expr = expr whileStmt ::= while (expr) stmt

4) “Single Production Terminator”• If there is single production

X ::=Y put an edge (X,Y) into graph• If there is a path from X to Z in the graph, and

there is rule Z ::= s1 s2 … sn then add rule

program ::= expr = expr | while (expr) stmt | stmt ; stmtSeq stmtSeq ::= expr = expr | while (expr) stmt | stmt ; stmtSeq stmt ::= expr = expr | while (expr) stmt

X ::= s1 s2 … snAt the end, remove all single productions.

5) No more than 2 symbols on RHS

stmt ::= while (expr) stmtbecomes

stmt ::= while stmt1

stmt1 ::= ( stmt2

stmt2 ::= expr stmt3

stmt3 ::= ) stmt

6) A non-terminal for each terminal

stmt ::= while (expr) stmtbecomes

stmt ::= Nwhile stmt1

stmt1 ::= N( stmt2

stmt2 ::= expr stmt3

stmt3 ::= N) stmtNwhile ::= whileN( ::= (N) ::= )

Parsing using CYK Algorithm

• Transform grammar into Chomsky Form:1. remove unproductive symbols2. remove unreachable symbols3. remove epsilons (no non-start nullable symbols)4. remove single non-terminal productions X::=Y5. transform productions of arity more than two6. make terminals occur alone on right-hand sideHave only rules X ::= Y Z, X ::= t, and possibly S ::= “”

• Apply CYK dynamic programming algorithm